Ada Lovelace and the Logic of the Analytical Engine

Ada Lovelace and the Logic of the Analytical Engine

Philip Welch, School of Mathematics. Annabella Milbanke 1792-1860 George Byron 1788-1824 Byron leaving England

George Cruickshank 1816 Augusta Ada Byron 1815-1852 Ada Byron, Age 17 Charles Babbage 1791-1871 Difference Engine No. 1 (portion) 1832 William King, Earl of Ockham, later Lord Lovelace 1805-1893 Byron in Abyssinian dress Ashley Combe, Somerset “ ‘Mathematics’ should be written on my jaw ”

as Lady King, (Age 20) by Margaret Carpenter, 1835 after a sketch by A.E. Chalon 1838. (Age 23) J.R. Jacquard 1752-1834 A Jacquard Loom

The Analytical Engine Punched Cards: Operator & Variable

Lovelace: ”The bounds of arithmetic were outstepped the moment of applying the cards had occured” And 150 years later ...

Luigi Federico Menabrea, 1805-1893 & later Prime Minister of Italy Menabrea’s article - 1842

He describes the Analytical Engine in great detail - not the would be • physical construction, but the methodology of the computations.

He gives three examples of calculations. We’ll look at that for a simple • (2 2) simultaneous equation with unknowns x and y: £

mx ny d + = m0x n0y d0 + = This will have solution:

dn0 d0n d0m dm0 x ° ; y ° = n m nm = mn m n 0 ° 0 0 ° 0

Bernoulli Numbers

These are a sequence of fractions B1,B3,B5,...B2n 1,... They crop up • + again and again in analysis and number theory. There are many ways to define them. We can define them through a power series • x x x2 x4 x6 ex 1 1 2 B1 2! B3 4! B5 6! ° = ° + + + +···

Or explicitly: m k m v k v B2m 1 ( 1) . + = k 0 v 0 ° √v!k 1 X= X= +

Lovelace chose: m 1 ° 2m 1 B2k 1 B2m 1 + + . + =°k 0 √ 2k 1 !2m 2k 1 X= + ° + Her choice looks like:

B2m 1 a1B1 a3B3 a2m 1B2m 1 + = + +···+ ° ° Rearranging: Rearranging:

B a B 3 = 1 1 B d B d B 5 = 1 1 + 3 3 B e B e B e B 7 = 1 1 + 3 3 + 5 5 So her programme to compute B proceeds by computing first B , then • 7 1 B3 from B1; then B5 from B3 and B1 etc.

This is known as a course-of-values recursion because it requires the • complete course of all the previous values to get the next one.

An (ordinary) recursion would just have defined B in terms of B , and B • 3 1 5 in terms of B3, B7 in terms of B5, that is using only the most immediate previous value etc.

B a B 3 = 1 1 B d0 B 5 = 3 3 B e0 B 7 = 5 5 Lovelace’s programme to compute Bernoulli Numbers

The DE could perform the following operations: The arithmetical operations , , are operations. œ + £ ° œ Any sequence of operations is an operation.

œ (Iteration) Any finite iteration of an operation is an operation.

(Today we should say that the DE could compute any primitively recursive function.) But the Analytical Engine additionally had:

œ Conditional iteration If P is an operation and T is a test on register contents, then the result of iterating P until T succeeds is an operation.

Was the Analytical Engine just a larger Difference Engine?

The Difference Engine, although it had the same arithmetical operations • as the Analytical Engine, was logically on a lower plane. (Today we should say that the DE could compute any primitively recursive function.) But the Analytical Engine additionally had:

œ Conditional iteration If P is an operation and T is a test on register contents, then the result of iterating P until T succeeds is an operation.

Was the Analytical Engine just a larger Difference Engine?

The Difference Engine, although it had the same arithmetical operations • as the Analytical Engine, was logically on a lower plane.

The DE could perform the following operations: The arithmetical operations , , are operations. œ + £ ° œ Any sequence of operations is an operation.

œ (Iteration) Any finite iteration of an operation is an operation. Was the Analytical Engine just a larger Difference Engine?

The Difference Engine, although it had the same arithmetical operations • as the Analytical Engine, was logically on a lower plane.

The DE could perform the following operations: The arithmetical operations , , are operations. œ + £ ° œ Any sequence of operations is an operation.

œ (Iteration) Any finite iteration of an operation is an operation.

(Today we should say that the DE could compute any primitively recursive function.) But the Analytical Engine additionally had:

œ Conditional iteration If P is an operation and T is a test on register contents, then the result of iterating P until T succeeds is an operation. Flowchart for a Shepherdson-Sturgis-Minsky Register Machine So the answer is “Yes”: the Analytical Engine, using Turing and Gödel’s ideas, can be seen to be universal.

Moreover as Turing showed, the programs could also be treated as data, and simple be stored inside the machine prior to operation.

Was the Analytical Engine ‘Universal’?

A universal computer is one equipped with a universal program that is • one that could theoretically emulate the action of every other program. Turing, defining the term, proved that his ‘machines’ were universal.

What Babbage and Lovelace lacked, was a coding method, a way of • coding up program instructions by numbers, which a universal machine could decode and then simulate.

Kurt Gödel (1930) gave a way of coding alphabets, then words, then • sentences for use in his famous Incompleteness Theorems. Turing then used this idea to code up programs (1936). Using this a Shep.-S-M Register machine only needs a half dozen Registers to be a fully universal machine. Was the Analytical Engine ‘Universal’?

A universal computer is one equipped with a universal program that is • one that could theoretically emulate the action of every other program. Turing, defining the term, proved that his ‘machines’ were universal.

What Babbage and Lovelace lacked, was a coding method, a way of • coding up program instructions by numbers, which a universal machine could decode and then simulate.

Kurt Gödel (1930) gave a way of coding alphabets, then words, then • sentences for use in his famous Incompleteness Theorems. Turing then used this idea to code up programs (1936). Using this a Shep.-S-M Register machine only needs a half dozen Registers to be a fully universal machine.

So the answer is “Yes”: the Analytical Engine, using Turing and Gödel’s ideas, can be seen to be universal.

Moreover as Turing showed, the programs could also be treated as data, and simple be stored inside the machine prior to operation. The vistas beyond

Neither Babbage, nor Lovelace knew of universality, but both saw potentialities beyond the calculations they programmed.

Babbage’s thesis: “These two memoirs [Menabrea’s and Lovelace’s articles] furnish a complete demonstration - that the whole of the development and operations of analysis are capable of being executed by machinery.” (Babbage’s italics.) Do machines think? “Lovelace’s Objection”

Lovelace: “It is desirable to guard against the possibility of exaggerated ideas that might arise as to the powers of the Analytical Engine [which] has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform . . . . but it has no power of anticipating any analytical relations or truths.”

This is discussed in a famous paper by Turing in the journal Mind on the • possibility of machines having intelligent thought. But again. . . .

Lovelace: “Again it might act upon things other than number, ... Supposing for instance , that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.” The effect on computing machinery on pure mathematics

Lovelace again: “It is however pretty evident, on general principles, that in devising for mathematical truths a new form in which to record and throw themselves out for actual use, views are likely to be induced, which should again react on the more theoretical phase of the subject”